Achieving desired throughput in Slotted ALOHA protocol

You have two computers, A and B, sharing a wireless network in your room. The network
runs the slotted Aloha protocol with equal-sized packets. You want B to get twice the throughput over the wireless network as A whenever both nodes are backlogged. You configure A to send packets with probability p. What should you set the transmission probability of B to, in order to achieve your throughput goal?

(A) p / 1+p
(B) p / 1+2p
(C) 2p / 1+p
(D) 1/2

Let B’s transmission probability be pb.

The utilization is equal to p(1−pb)+(1−p)pb.

A’s throughput is the first term, p(1 − pb),
and B’s throughput is the second term, pb(1 − p).

Using the fact that we want B to get twice the throughput of A, and solving for pb in terms of p,

2. p(1 − pb) = pb(1 − p)
solving, pb = 2p/1+p